Anharmonic interaction shift

The quantum theory of vibrational relaxation in low-temperature ordered solids is well develojjed, at least for weak interactions. Starting from the harmonic solid, with known normal mode energies , the anharmonic interactions between modes are introduced as an ordered perturbation and the renormalized mode energies are calculated, usually by temperature Green s function methods, for each order of jierturbation. The calculated energy shifts j — are complex. 

The sensitivity of the phonon frequencies to temperature shows quite clearly the importance of their anharmonicity.42 The width of the Raman peaks, very small at low temperature ( 1cm-1), evolves in parallel with the frequency shift with temperature, which is still a consequence of the phonon-phonon interactions due to the anharmonicity. The fundamental reason for this strong anharmonicity, as well as the importance of the equilibrium-position shifts between 4 and 300 K,45 resides in the weakness of the van der Waals cohesive forces in the molecular crystal. 

The phase relation of the excitation piezo and the vibration of the cantilever in time yields often highly useful information on the interaction of tip and sample surface (Fig. 2d) (49). The phase angle shift A( between excitation oscillation and forced oscillation of the cantilever can be described by a simple harmonic (50-52), but also more complicated anharmonic (19), treatment of a forced oscillator. The phase angle shift can be used to obtain contrast due to different surface characteristics related to materials properties (intermittent contact mode SFMpftase imaging) (49). These properties include, depending on the operation conditions, differences in adhesion (53) or differences in Yoimg s modulus (50-52,54,55). Hence the amorphous and crystalline phases in semiciystalline polymers can be clearly differentiated, as well as different phases in polymer blends or filled systems. 

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